% This is the quiz 2 for Stat 6502

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\centerline{\bf{CALIFORNIA STATE UNIVERSITY, EAST BAY }}
\centerline{\bf{STATISTICS DEPARTMENT}} \centerline{ }
\centerline{\bf{Statistics 6502 Mathematical Statistics}}
\centerline{\bf{Spring 2013}}

\medskip
\noindent \centerline{\bf{Quiz II}}

\begin{enumerate}

% *******************************************************************
\item
{\bf ( 40 points )} You are given a single observation from either
the standard normal or the uniform distribution on $(-4, 4)$
distribution.
\begin{enumerate}
\item
Accurately sketch a picture of the null and alternative densities
on one set of axes.
\item
Find the most powerful test of size $\alpha = 0.05$ for
\begin{eqnarray*}
  H_0: X \sim Normal(0,1^2)
\end{eqnarray*}
versus
\begin{eqnarray*}
  H_1: X \sim Uniform(-4,4).
\end{eqnarray*}
\item
Calculate the power, $\pi$, of the test.
\item
Calculate the probability of Type II Error, $\beta$, for the the test.
\end{enumerate}

% *******************************************************************
\item
% problem:
% source Casella and Berger, Problem 8.23
{\bf (Extra Credit)}
Let $X$ be a random variable whose p.m.f. under $H_0$ and $H_1$ is given by

\begin{center}
\begin{tabular}{c|ccccccc}

  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  $x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
  \hline
  $f_0(x)$ & .01 & .01 & .01 & .01 & .01 & .01 & .94 \\
  $f_1(x)$ & .06 & .05 & .04 & .03 & .02 & .01 & .79 \\
  \hline
\end{tabular}
\end{center}

\begin{enumerate}
\item Accurately sketch a picture of the densities on one set of
axes.
\item Use the Neyman-Person Lemma to find the most powerful test for
$H_0$ versus $H_1$ with size $\alpha = .04$.
\item Compute the probability of Type II Error for this test.
\end{enumerate}

% *******************************************************************



\end{enumerate}

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